CSCI 1951-G Optimization Methods in Finance Part 10: Conic Optimization

CSCI 1951-G Optimization Methods in Finance Part 10: Conic Optimization April 6, 2018 1 / 34

This material is covered in the textbook, Chapters 9 and 10. Some of the materials are taken from it. Some of the figures are from S. Boyd, L. Vandenberge s book Convex Optimization https://web.stanford.edu/~boyd/cvxbook/. 2 / 34

Outline 1. Cones and conic optimization 2. Converting quadratic constraints into cone constraints 3. Benchmark-relative portfolio optimization 4. Semidefinite programming 5. Approximating covariance matrices 3 / 34

Cones A set C is a cone if for every x C and θ 0, θx C Example: {(x, x ), x R} R 2 Is this set convex? 4 / 34

Convex Cones A set C is a convex cone if, for every x 1, x 2 C and θ 1, θ 2 0, Example: θ 1 x 1 + θ 2 x 2 C. x 1 x 2 0 Figure 2.4 The pie slice shows all points of the form θ 1x 1 + θ 2x 2,where θ 1, θ 2 0. The apex of the slice (which corresponds to θ 1 = θ 2 = 0) is at 0; its edges (which correspond to θ 1 = 0 or θ 2 = 0) pass through the points x 1 and x 2. 5 / 34

Conic optimization Conic optimization problem in standard form: min c T x Ax = b x C where C is a convex cone in finite-dimensional vector space X. Note: linear objective function, linear constraints. If X = R n and C = R n +, then...we get an LP! Conic optimization is a unifying framework for linear programming, second-order cone programming (SOCP), semidefinite programming (SDP). 6 / 34

Norm cones Let be any norm in R n 1. The norm cone associated to is the set C = {x = (x 1,..., x n ) : x 1 (x 2,..., x n ) } It is a convex set. 7 / 34

Second-order cone in R 3 The second-order cone is the norm cone for the Euclidean norm 2. 1 t 0.5 0 1 0 x 2 1 1 x 1 0 1 Figure 2.10 Boundary of second-order cone in R 3, {(x 1,x 2,t) (x 2 1+x 2 2) 1/2 t}. What happens when we slice the second-order cone? I.e., when we take the intersection with a hyperplane? We obtain ellipsoidal sets. 8 / 34

Rewriting constraints Let s rewrite C = {x = (x 1,..., x n ) : x 1 (x 2,..., x n ) 2 } as x 1 0, x 2 1 x 2 2 x 2 n 0 This is a combination of an linear and a quadratic constraints. Also: convex quadratic constraints can be expressed as second-order cone membership constraints. 9 / 34

Rewriting constraints Quadratic constraint: x T P x + 2q T x + γ 0 Assume P w.l.o.g. positive definite, so the constraint is...convex. Also assume, for technical reasons, that q T P q γ 0. Goal: rewrite the above constraint as a combination of linear and second-order cone membership constraints. 10 / 34

Rewriting constraints Because P is positive definitive, it has a Cholesky decomposition: invertible R s.t. P = RR T. Rewrite the constraint as: (R T x) T (R T x) + 2q T x + γ 0 Let y = (y 1,..., y n ) T = R T x + R 1 q The above is a bijection between x and y. We are going to rewrite the constraint as a constraint on y. 11 / 34

Rewriting constraints The constraint: It holds (R T x) T (R T x) + 2q T x + γ 0 y T y = (R T x) T (R T x) + 2q T x + q T P 1 q Since there is a bijection between y and x, the constraint can be satisfied if and only if y s.t. y = R T x + R 1 q, y T y q T P q γ 12 / 34

The constraint is equivalent to: Rewriting constraints y s.t. y = R T x + R 1 q, y T y q T P q γ Lets denote with y 0 the square root of the r.h.s. of the right inequality: y 0 = q T P q γ R + Consider the vector (y 0, y 1,..., y }{{ n ). } y The right inequality then is y 2 0 y T y = n i=1 Taking the square root on both sides: y 0 n yi 2 = y 2 i=1 This is the membership constraint for the second-order cone in R n+1. y 2 i 13 / 34

Rewriting constraints We rewrite the convex quadratic constraint as x T P x + 2q T x + γ 0 (y 1,..., y n ) T = R T x + R 1 q (y 0, y 1,..., y n ) C y 0 = q T P q γ R + which is a combination of linear and second-order cone membership constraints. 14 / 34

Benchmark-relative portfolio optimization Given a benchmark strategy x B (e.g., an index) develop a portfolio x that tracks x B, but adds value by beating it. I.e., we want a portfolio x with positive expected excess return: µ T (x x B ) 0 and specifically want to maximize the expected excess return. Challenge: balance expected excess return with its variance. 16 / 34

Tracking error and volatility constraints The (predicted) tracking error of the portfolio x is TE(x) = (x x B ) T Σ(x x B ) It measure the variability of excess returns. In benchmark-relative portfolio optimization, we solve mean-variance optimization w.r.t. the expected excess return and tracking error: max µ T (x x B ) (x x B ) T Σ(x x B ) T 2 Ax = b 17 / 34

Comparison with mean-variance optimization We have seen MVO as: min 1 2 xt Σx µ T x R Ax = b or max µ T x δ 2 Σx Ax = b How do they differ from max µ T (x x B ) (x x B ) T Σ(x x B ) T 2 Ax = b The latter is not a standard quadratic program: it has a nonlinear constraint. 18 / 34

max µ T (x x B ) (x x B ) T Σ(x x B ) T 2 Ax = b The nonlinear constraint is...convex quadratic We can rewrite it as a combination of linear and second-order cone membership, and solve the resulting convex conic problem. 19 / 34

SemiDefinite Programming (SDP) The variables are the entries of a symmetric matrix in the cone of positive semidefinite matrices. 1 z 0.5 0 1 0 y 1 0 x 0.5 1 Figure 2.12 Boundary of positive semidefinite cone in S 2. 21 / 34

Application: approximating covariance matrices Portfolio Optimization almost always requires covariance matrices. These are not directly available, but are estimated. Estimation of covariance matrices is a very challenging task, mathematically and computationally, because the matrices must satisfy various properties (e.g., symmetry, positive semidefiniteness). To be efficient, many estimation methods do not impose problem-dependent constraints. Typically, one is interested in finding the smallest distortion of the original estimate that satisfies the desired constraints; 22 / 34

Application: approximating covariance matrices Let ˆΣ S n be an estimate of a covariance matrix ˆΣ is symmetric ( S n ) but not positive semidefinite. Goal: find the positive semidefinite matrix that is closest to ˆΣ w.r.t. the Frobenius norm: d F (Σ, ˆΣ) = (Σ ij ˆΣ ij ) 2 i,j Formally: nearest covariance matrix problem: min Σ d F (Σ, ˆΣ) Σ C n s where C n s is the cone of n n symmetric and positive semidefinite matrices. 23 / 34

Application: approximating covariance matrices min Σ d F (Σ, ˆΣ) Σ C n s Introduce a dummy variable t and rewrite the problem as min t d F (Σ, ˆΣ) t Σ C n s The first constraint can be written as a second-order cone constraint, so the problem is transformed into a conic optimization problem. 24 / 34

Application: approximating covariance matrices Variation of the problem with additional linear constraints: Let E {(i, j) : 1 i n} Let (l ij, u ij ), for (i, j) E be lower/upper bounds to impose to the entries. We want to solve: min Σ d F (Σ, ˆΣ) l ij < Σ ij < u ij, (i, j) E Σ C n s 25 / 34

Application: approximating covariance matrices For example, let ˆΣ be an estimation of a correlation matrix. Correlation matrix have all diagonal entries equal to 1. We want to solve the nearest correlation matrix problem. We choose E = {(i, i), 1 i n}, l i = 1 = u i, 1 i n 26 / 34

Application: approximating covariance matrices Many other variants are possible: Force some entries of ˆΣ to remain the same in Σ; Weight the changes to different entries differently, because we trust some more than other; Impose lower bounds to the minimum eigenvalue of Σ, to reduce instability; All of these can be easily solved with SDP software. 27 / 34

A different point of view to SDP A n n matrix A is positive semidefinite if there are vectors x i,... x j such that A ij = x T i x j. We can then write a semidefinite program as a program involving only linear combinations of the inner products of the vectors x i,... x j : min i,j [n] i,j [n] c ij x T i x j a ijk x T i x j b j, k This form is particularly useful to develop approximation algorithms. 29 / 34

The MaxCut problem Given a graph G = (V, E), output a 2-partition of V so as to maximize the number of edges crossing from one side to the other. Integer quadratic program: max (i,j) E 1 v i v j 2 v i { 1, 1}, 1 i n The decision version of the problem is NP-complete. 30 / 34

The MaxCut problem Steps for an approximation algorithm for MaxCut: 1 Relax the original problem to an SDP; 2 Solve the SDP; 3 Round the SDP solution to obtain an integer solution to the original problem. 31 / 34

The MaxCut problem Integer quadratic program: max (i,j) E 1 v i v j 2 v i { 1, 1}, 1 i n SDP Relaxation: max (i,j) E 1 v T i v j 2 v i 2 2 1, 1 i n v i R n It is a relaxation: the optimal obj. value will be larger than the one for the original problem. 32 / 34

The MaxCut problem max (i,j) E 1 v T i v j 2 v i 2 2 1, 1 i n v i R n The optimal solution is a set of unit vectors in R n To obtain a solution for the original problem, we need to round this solution and assign each vector to one value in { 1, 1}. Goemans and Willamson 1995: choose a random hyperplane that goes through the origin, and split the vectors depending on the side of the hyperplane. Approximation ratio: 0.87856 - ε (essentially optimal) 33 / 34